#8630899

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#57581	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(2,3)(7,9)(8,10)(14,15)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,19,23,22)(18,24,21,26)(20,29,30,27)(25,32,28,31)(34,40)(35,36)(42,47)(43,44);(4,11)(7,9)(12,16)(14,15)(18,21)(24,26)(25,28)(31,32)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20,23,30)(18,25,21,28)(19,27,22,29)(24,31,26,32)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,18)(19,24,22,26)(20,28,30,25)(27,31,29,32)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,19)(18,24,21,26)(20,27,30,29)(25,32,28,31)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538095);